In this work, we propose an architecture for measurement-based quantum computing that possesses the advantage of CV-based schemes and yet is compatible with probabilistic GKP qubit sources.
We consider a hybrid CV cluster state where each mode is substituted with a GKP qubit at random and with probability \((1-p_0)\) —or said the other way, a qubit cluster state where each node is substituted with a squeezed state with swap-out probability \(p_0\) .
The precise state we consider is the lattice from the Raussendorf, Harrington, Goyal (RHG) model [1]}, [2]}, [3]}, but our scheme can readily accommodate other error-correcting codes.
Our use of CV resources affords us an important alternative over existing approaches, wherein a qubit that failed to be produced must be erased from the lattice.
Instead, we replace the no-show qubit with a squeezed vacuum state: it can still encode logical information (albeit not as well as a GKP state [4]}) but has the distinction of being Gaussian and thus easily producible We cannot replace all modes with squeezed vacua because that would negate the error-correction benefits of the encoded qubits [5]}..
This approach – one of the main innovations in this work – propels us beyond existing fault tolerance methods, such as those that rely on lattice renormalization to deal with defects [6]}, [7]}, [8]}, [9]}.
To characterize the robustness of our architecture as a function of \(p_0\) , we perform Monte Carlo simulations of our architecture operating as a quantum memory.
We observe a minimum required squeezing of \({10.5}{}\)  or a maximum tolerable swap-out probability of \(p_0 \approx 0.236\) ; for an experimentally accessible squeezing value of \({15}{}\)  [10]}, our simulations suggest a swap-out threshold of \(p_0 \approx 0.133\) , which translates to substantially reduced multiplexing requirements for GKP generation.
In part these result stem from a tailored decoding procedures that we present and which allow us to perform fault-tolerant computation on our hybrid resource state.
